Preface

The main text of this article is reproduced by kind permission from:
The Mathematical Gazette80, November 1996, p.p. 466-475.

Some changes have been made from the original as follows:

Errors corrected in the coordinates of the sphenoid hendecahedron vertices A and L.

The dodecahemioctahedron renamed as the rhombic
dodecahemioctahedron, in order to distinguish it from the topologically similar
polyhedron with regular faces. Correction to its description - the hexagons are not faces.

The postscript on duality, cut from the original due to lack of
space, has been restored and updated.

Web links, etc. updated.

Change history

10 January 2010.Links to Maurice Starck's page updated.18 October 2006.Printable nets & link to Maurice Starck's page added.7 October 2006.Correction to errors introduced at last update. Preface added. Minor tidy-up.8 January 2006.Corrections and updates to original.10 June 2007.More corrections to corrections, etc.

Five space-filling polyhedra

Introduction

Solid shapes which pack together to fill space cover a large and varied
range. I recently found five such polyhedra which I have not seen
described elsewhere. The names which I have used for them here are rather
ferocious I am afraid, but that just happens to be the way the names of
polyhedra work.

A polyhedron with eleven faces is called a hendecahedron, from the Greek
for eleven. The one shown in Figure 1 has two planes of symmetry, i.e. it
is bisymmetric. This hendecahedron also has eleven vertices; polyhedra
with the same number of faces as vertices are not very common. It has 2
large rhombic faces, a small rhombic face (which in the proportions used
here is square), 4 congruent iscosceles triangular faces which meet along
edges at right angles, and 4 congruent kite-shaped faces. (see
Appendix B for coordinates.)

Figures 2 and 3 shows how four hendecahedra together form a kind of
hexagonal boat shape which will stack in interlocking layers. This boat
shape is also a 'translation unit' - it can be regularly stacked in a
lattice to fill space, without any rotation or reflection. This lattice is
similar to the body-centred cubic, but scaled vertically by a factor which
is here one-half (but see below). In Figure 4 the way the hendecahedra
themselves stack together to form a space-filling 'honeycomb' can be seen.

The particular polyhedron described here has an arbitrary height (chosen
for convenience of its coordinates): it can be distorted vertically by
stretching, to give an infinite series of shapes which are all
space-fillers. In Figure 4, vertical lines can be seen running through the
stack. Figure 2 shows how these lines are made from the edges where two
triangular faces meet. In Figure 3, the lines are viewed end-on and appear
as the corners where four hendecahedra from each layer meet. It is obvious
from this that the angle between two triangular faces must be a right
angle, but what should the other angles seen in Figure 3 be? These other
angles can be varied by rotating the triangular faces together about their
vertical edges, whilst maintaining the right angles and the overall
symmetry of the shapes, up to the points at which faces merge or
disappear. This rotation generates space-filling hendecahedra varying
continuously from ones with broad front rhombs and blunt backs to ones
with narrow front rhombs and sharper backs. The two distortions together
yield a doubly-varying range of space-fillers.

By cutting a hendecahedron in half horizontally and inserting a
(pentagonal) prismatic centre section, an elongated bisymmetric
hendecahedron is formed (Figure 5). The square face has become hexagonal,
and the triangles are now trapezia (the term trapezium has two
quite different interpretations - I use the term to describe a
quadrilateral with one pair of parallel sides). The new solid has fourteen
vertices with coordinates obtained from the original solid as described in
Appendix B.

The shape fills space in a similar way to the unstretched variant. It
may be distorted vertically in two independent zones: one being the
prismatic centre zone and the other the tapering top and bottom ends
corresponding to an unelongated shape. Together with rotation of the
trapezoidal faces, this yields a threefold range of distortions which
still fill space.

The sphenoid hendecahedra

'Sphenoid' means wedge-shaped, which is an apt description for the
hendecahedron shown in Figure 6. This also has eleven vertices, but it has
only one plane of symmetry: top and bottom halves have reflective
symmetry, but left and right halves are different, as can be seen in the
end views. The sphenoid hendecahedron has three sizes of kite-shaped face
and two types of isosceles triangle, all coming in pairs. In the
proportions used here, the rhombic face is square. Unexpectedly the
numbers of 3- and 4- sided faces are the same as for the bisymmetric
variant, and indeed both hendecahedra have the same topology - they can be
simply distorted into one another.

Figure 7 shows how six units pack like flower petals to form a 'floret'.
A second floret fits up to it from below and is the other way up. The two
florets form a translation unit which packs in a simple hexagonal lattice,
here with a height equal to the unit length.

Florets will stack in layers, alternate ways up, to form faceted
columns. In Figure 8, one floret is laid upon another, and in Figure 9 it
can be seen how the florets also fit together side by side to form a
layer. The layer is not quite symmetrical, alternate layers being right-
and left-handed about the junction of three florets. Perhaps a little
harder to visualise is the way the whole structure of layers and columns
interlocks with no gaps, as illustrated in Figure 10.

A new polyhedron can also be made by elongation (Figure 11): in the
elongated sphenoid hendecahedron, the square face again becomes a hexagon
and the triangles become trapezia. It also has fourteen vertices, with
coordinates obtained from the original solid as described in Appendix B, and fills space in a similar way to its unstretched cousin.

The two sphenoid shapes can be distorted vertically in the same way as
the bisymmetric ones, but have no analogue of the rotational distortion.

The rhombic dodecahemioctahedron

There are a number of solids with faces which pass through their centre,
so that the face has no inside or outside but can be seen from both sides
in different places. These central faces are typically parallel to the
faces of some normal convex polyhedron, but number half as many. Figure 12
shows two of the four hexagonal face planes corresponding to half an octahedron which make
this solid 'hemi-octa.' The hemi faces are the visible parts of the hexagons, and
comprise 'butterfly' cross-quadrilaterals. Neither the hexagons nor the octahedron are regular, the
octahedron being slightly flattened or 'oblate'. The solid also has 4
rhombic faces and 8 (isosceles) triangular ones, making in all 12 ordinary
faces. Hence the name rhombic dodecahemioctahedron.

Figures 13 and 14 illustrate two other ways of deriving its shape:
Figure 13 as a cube cut into six square pyramids meeting at
the centre, with two opposing pyramids removed and four new pyramids stuck
onto the square bases of the remaining ones, and Figure 14 as four oblate octahedra joined face
to face around a central vertex. It can also be thought of as
a rhombic dodecahedron with two oblate octahedra removed leaving dimples
behind.

The corner of one rhombic dodecahemioctahedron exactly fits the dimple
of another (Figure 15). A series of units can be fitted together in two
basic ways, as in Figure 16. In various combinations, these give rise to
several different packings which fill space. These packings are not true
lattices, since they have 'false' edges, where the edge of one or more
units lies across the face of another.

The most regular packing I have found is shown in Figure 17. The units
form layers, each of which has alternate rows of peaks and dimples. The
packing does not have full cubic symmetry, since the rows of peaks and
dimples give each layer a directional 'grain'. But there is no distinction
between the three main axes. Another packing, shown in Figure 18, forms
distinct layers of peaks alternating with layers of dimples. This pattern
is only seen in one plane, so the packing has a definite way up or
orientation.

The rhombic dodecahemioctahedron is pristine, which means it cannot be
distorted in any way and still fill space.

Postscript on duality

The reciprocal or dual of a polyhedron is a kind of anti-twin. It has
vertices corresponding to the faces of the original, and faces to the
vertices. It has the same number of edges as the original, but at
right-angles to them. The relationship is reciprocal, which means that the dual of
a dual is the original shape.

Self-dual polyhedra are possible: the regular tetrahedron is an example.
Such a shape must have the same number of vertices as faces, and without
going into detail the pattern, or topology, of edges around every face must corresond
exactly to the pattern of edges meeting at every vertex. The two
unelongated hendecahedra described share the same topology, which fulfils this requirement.
But neither meets more subtle requirements for the angles of faces, etc. A self-dual
hendecahedron does exist - it is the canonical form of this topology and is not a
space-filler.

The rhombic dodecahemioctahedron does not have a finite dual. The dual
feature corresponding to a face through the centre would be a vertex at
infinity, in both opposing directions orthogonal to the face, therefore
joining finite edges to it is not possible.

Acknowledgements

The 3-D views of polyhedra were created by WimpPoly and Polydraw
software for the RISC OS (Acorn) operating system, from Fortran
Friends, PO Box 64, Didcot, Oxon OX11 0TH. I am indebted to the
author K. M. Crennell for advice, enthusiasm and pre-release software.